In a circle, there are two chords opposite to the centre of the circle. The length of one chord and the radius of the circle is 6 cm and 5 cm respectively, and the distance between both the chords is 7 cm. Find the length of the second chord.

Options

A. a) 8 cm
B. b) 10 cm
C. c) 6 cm
D. d) 14 cm

Answer

To solve the problem step by step, we will use the properties of circles and right triangles.

### Step 1: Understand the Given Information
We have a circle with:
- One chord (AB) of length 6 cm.
- The radius (r) of the circle is 5 cm.
- The distance between the two chords (AB and CD) is 7 cm.

### Step 2: Draw the Circle and Chords
Draw a circle and label the center as O. Draw the chord AB and mark its length as 6 cm. Draw the second chord CD, which is parallel to AB, and the distance between the two chords is 7 cm.

### Step 3: Find the Distance from the Center to Chord AB
Since the chord AB is 6 cm long, the perpendicular distance from the center O to the chord AB (let's call this distance OE) can be found using the Pythagorean theorem. The half-length of the chord AB is 3 cm (since 6 cm / 2 = 3 cm).

Using the Pythagorean theorem:
\[ OE^2 + (AB/2)^2 = r^2 \]
\[ OE^2 + 3^2 = 5^2 \]
\[ OE^2 + 9 = 25 \]
\[ OE^2 = 16 \]
\[ OE = 4 \text{ cm} \]

### Step 4: Find the Distance from the Center to Chord CD
Since the distance between the two chords is 7 cm, we can find the distance OF from the center O to the chord CD. The distance EF between the two chords can be expressed as:
\[ EF = OE + OF \]
Given that EF = 7 cm and OE = 4 cm:
\[ 7 = 4 + OF \]
\[ OF = 3 \text{ cm} \]

### Step 5: Find the Length of Chord CD
Now, we can find the length of the second chord CD using the distance OF. The distance OF is the perpendicular distance from the center O to the chord CD.

Using the Pythagorean theorem again:
\[ OF^2 + (CD/2)^2 = r^2 \]
Substituting the known values:
\[ 3^2 + (CD/2)^2 = 5^2 \]
\[ 9 + (CD/2)^2 = 25 \]
\[ (CD/2)^2 = 16 \]
\[ CD/2 = 4 \]
Thus, the length of chord CD is:
\[ CD = 2 \times 4 = 8 \text{ cm} \]

### Conclusion
The length of the second chord CD is 8 cm.

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